Spectral Treatment of Distributed-Order Caputo Models: Applications to Nonlinear Fractional Sine and Klein-Gordon Differential Equations (Contemporary Mathematics, 2025)
In this study, we present a highly efficient spectral numerical approach for solving nonlinear Fractional Distributed-Order Sine-Gordon Differential Equations (FDO-SGDEs) and Fractional Distributed-Order Klein-Gordon Differential Equations (FDO-KGDEs) considering initial and Dirichlet boundary conditions. Our proposed method utilizes the fractional-order shifted Legendre-Gauss collocation method and shifted Chebyshev-Gauss collocation method, leveraging the Riemann-Liouville fractional derivative to transform the given problems into systems of algebraic equations. The effectiveness and accuracy of this technique are demonstrated through the successful resolution of four illustrative examples.
Keywords: fractional Klein-Gordon models, fractional shifted Legendre and Chebyshev polynomials, Caputo fractional, collocation method distributed fractional